feat: Add results related to MSSM

HepLean/AnomalyCancellation/MSSMNu/Y3.lean

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+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license.
+Authors: Joseph Tooby-Smith
+-/
+import HepLean.AnomalyCancellation.MSSMNu.Basic
+/-!
+# The definition of the solution Y₃ and properties thereof
+
+We define $Y_3$ and show that it is a double point of the cubic.
+
+# References
+
+The main reference for the material in this file is:
+
+- https://arxiv.org/pdf/2107.07926.pdf
+
+-/
+
+universe v u
+
+namespace MSSMACC
+open MSSMCharges
+open MSSMACCs
+open BigOperators
+
+/-- $Y_3$ is the charge which is hypercharge in all families, but with the third
+family of the opposite sign. -/
+def Y₃AsCharge : MSSMACC.charges := toSpecies.symm
+  ⟨fun s => fun i =>
+    match s, i with
+    | 0, 0 => 1
+    | 0, 1 => 1
+    | 0, 2 => - 1
+    | 1, 0 => -4
+    | 1, 1 => -4
+    | 1, 2 => 4
+    | 2, 0 => 2
+    | 2, 1 => 2
+    | 2, 2 => - 2
+    | 3, 0 => -3
+    | 3, 1 => -3
+    | 3, 2 => 3
+    | 4, 0 => 6
+    | 4, 1 => 6
+    | 4, 2 => - 6
+    | 5, 0 => 0
+    | 5, 1 => 0
+    | 5, 2 => 0,
+  fun s =>
+    match s with
+    | 0 => -3
+    | 1 => 3⟩
+
+/-- $Y_3$ as a solution. -/
+def Y₃ : MSSMACC.Sols :=
+  MSSMACC.AnomalyFreeMk Y₃AsCharge (by rfl) (by rfl) (by rfl) (by rfl) (by rfl) (by rfl)
+
+lemma Y₃_val : Y₃.val = Y₃AsCharge := by
+  rfl
+
+lemma doublePoint_Y₃_Y₃ (R : MSSMACC.LinSols) :
+    cubeTriLin (Y₃.val, Y₃.val, R.val) = 0 := by
+  rw [← TriLinearSymm.toFun_eq_coe]
+  simp only [cubeTriLin, cubeTriLinToFun, MSSMSpecies_numberCharges]
+  rw [Fin.sum_univ_three]
+  rw [Y₃_val]
+  rw [Y₃AsCharge]
+  repeat rw [toSMSpecies_toSpecies_inv]
+  rw [Hd_toSpecies_inv, Hu_toSpecies_inv]
+  simp
+  have hLin := R.linearSol
+  simp at hLin
+  have h3 := hLin 3
+  simp [Fin.sum_univ_three] at h3
+  linear_combination 6 * h3
+
+end MSSMACC